Cosgrove's name was not on the initial list of possible nominees that surfaced last week after Shinseki resigned amid an ongoing scandal involving egregiously long waits and secret waitlists for veterans to get care at VA facilities around the country.

Shinseki's replacement was widely expected to be someone with either military or VA experience. Previously mentioned possibilities include Sloan Gibson, Shinseki's interim replacement; former Sen. Jim Webb (D-Va.); Army Gen. Peter Chiarelli; Rep. Tammy Duckworth (D-Ill.); and Army Chief of Staff Ray Odierno.

The White House did not immediately respond to a request for comment.

The Cleveland Clinic operates 11 hospitals—10 in the Cleveland area and one in Weston, Fla.—with $6.5 billion in 2013 revenue. The organization is a small fraction of the size of the VA healthcare system, which has a $55.9 billion budget and 151 hospitals among 1,700 facilities across the country. The VA serves nearly 8.8 million vets and qualifying family members.

Cosgrove, a cardiac surgeon and Vietnam veteran who was awarded the Bronze Star, first joined the Cleveland Clinic in 1975. For the past decade, he has served at the helm of the system, which is often recognized for its quality of patient care and patient experience.

Although Cosgrove's executive experience is with a much smaller healthcare system, it's one that has drawn praise from President Barack Obama, who is looking for a leader with the expertise to fix the problems with the VA and also enough stature to stanch the political damage from the controversy.

In a 2012 presidential debate, Obama called the Cleveland Clinic “one of the best healthcare systems in the world.” Last year, Cosgrove published “The Cleveland Clinic Way,” a book that argued why the organization should serve as a model of healthcare for the U.S.